Thursday, November 28, 2019

trigonometry - Euler's formula simplification



I have been trying to simplify this expression:



$$\cos (\theta) - \frac{1}{\cos (\theta) + i \sin(\theta)}$$




into:



$$\ i \sin(\theta).$$



However I can't find what steps to take to get to this simplification.



Euler's formula states:



$$\ e^{i\theta}= \cos (\theta) + i \sin(\theta) $$




It is linked to this formula however I am not sure how to go about this.


Answer



$$cos(\theta)-\frac{1}{\cos(\theta)+i\sin(\theta)}=cos(\theta)-\frac{1}{e^{i\theta}}=$$
$$cos(\theta)-e^{-i\theta}=\frac{1}{2}e^{i\theta}+\frac{1}{2}e^{-i\theta}-e^{-i\theta}=$$
$$=\frac{1}{2}e^{i\theta}-\frac{1}{2}e^{-i\theta}=i\Big(\frac{1}{2i}e^{i\theta}-\frac{1}{2i}e^{-i\theta}\Big)=i\sin(\theta)$$


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